Free vibration analysis of rectangular plates with central cutout

In this paper an examination of methods for vibration analysis of moderately thick rectangular plates has been presented. First, the state-of-the art in thick plate vibration theories and analysis methods is described and it is followed with basic equations of the original Mindlin theory, which represents a starting point for the development of all other mathematical models. Then, the problem of analytical solving of equilibrium equations is considered based on the modified Mindlin theory of thick plate vibrations, which has been published in the literature recently. Further, energy methods that can be applied to arbitrary boundary conditions are discussed and outline of the assumed mode method is presented. Finally, in the context of numerical methods a new quadrilateral finite element, based on the above mentioned advanced thick plate theory, is included. It should be emphasized that it doesn’t suffers of shear locking problem associated with finite elements, due to natural relation among bending and shear polynomials, and moreover, it gives very accurate results. Application of the presented methods is illustrated by several numerical examples which include natural vibration analyses of rectangular plates with various thicknesses and different combinations of boundary conditions (simply supported, clamped, free and elastically restrained). Comparisons of natural frequencies and mode shapes with results available in the relevant literature and with those obtained by the commercial finite element software are also provided.

This article presents a simple and efficient procedure for the natural vibration analysis of rectangular plates and stiffened panels in contact with fluid on one side. The assumed mode method is applied, where the natural frequencies and mode shapes are obtained by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equation of motion. The Mindlin thick plate theory is applied for a plate, and in the case of stiffened panels, the effect of framing is taken into account by adding its strain and kinetic energies to the corresponding plate energies. Potential flow theory assumptions are adopted for the fluid, and free surface waves are ignored. The fluid velocity potential is derived from the boundary conditions for the fluid and structure and is utilized for the calculation of added mass using the assumed modes. The applicability and accuracy of the developed procedure are illustrated with several numerical examples using a developed in-house code. A comparison of the results with those obtained by general purpose finite element analysis software is provided, where very good agreement is achieved.

Three-dimensional vibration solutions are presented for rectangular plates with mixed boundary conditions, based on the small strain linear elasticity theory. The analysis is focused on two kinds of rectangular plates, the boundaries of which are partially fixed while the others are free. One of those studied is a rectangular plate with partially fixed boundaries symmetrically arranged around four corners and the other one is a rectangular plate with partially fixed boundaries around one corner only. A global analysis approach is developed. The Ritz method is applied to derive the governing eigenvalue equation by minimizing the energy functional of the plate. The admissible functions for all displacement components are taken as a product of a characteristic boundary function and the triplicate Chebyshev polynomial series defined in the plate domain. The characteristic boundary functions are composed of a product of four components of which each corresponds to one edge of the plate. The R-function method is applied to construct the characteristic boundary function components for the edges with mixed boundary conditions. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. The influence of the length of the fixed boundaries and the plate thickness on frequency parameters of square plates has been studied in detail. Some valuable results are given in the form of tables and figures, which can serve as the benchmark for the further research.

An exact three dimensional solution for bending of thick rectangular FGM plate

Performance of similitude methods for structural vibration analyses of rectangular plates

Several sets of static beam functions have been established by the authors, which are the static solutions of beams under arbitrary static loads. The beam is a strip taken from the rectangular plate to be considered, and the static load acting on the beam is expanded into a series which may be a series of point loads or a series of sinusoid loads or a series of polynomial loads. These sets of static beam functions have been successfully applied to the vibration analysis of rectangular plates with varying complexity by the Rayleigh-Ritz method and finite strip method. Some numerical results are given. By comparing the results with those available from literature it can be shown that the present method is extremely effective for solving the aforementioned problems and usually a few static beam functions will be sufficient to give acceptable results.

The purpose of this paper is to study shear locking-free parametric earthquake analysis of thick and thin plates resting on Winkler foundation using Mindlin’s theory, to determine the effects of the thickness/span ratio, the aspect ratio and the boundary conditions on the linear responses of thick and thin plates subjected to earthquake excitations. In the analysis, finite element method is used for spatial integration and the Newmark-β method is used for the time integration. Finite element formulation of the equations of the thick plate theory is derived by using higher order displacement shape functions. A computer program using finite element method is coded in C++ to analyze the plates clamped or simply supported along all four edges. In the analysis, 8-noded finite element is used. Graphs are presented that should help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that 8-noded finite element can be effectively used in the earthquake analysis of thick plates. It is also concluded that, in general, the changes in the thickness/span ratio are more effective on the maximum responses considered in this study than the changes in the aspect ratio.

Introduction. The theory of plates with shear deformation has been developed for several decades beginning from the studies [13, 17] (first-order shear deformation theory or Reissner–Mindlin theory [2–4, 20]). The dynamics of plates with shear compliance and rotary inertia was for the first time studied in [6]. Later, extensive studies in this area were performed in [1, 5, 8, 10–12, 16, etc.], where the effect of shear compliance on the natural frequencies was numerically analyzed in detail using various theories. The following circumstance is noteworthy. The first studies on the dynamics of plates and rods with shear deformation ([6], etc.) discovered two spectra of natural frequencies (two very different (low and high) natural frequencies correspond to the same total deflection mode shape). However, the sense of these two branches is still unclear. It is usually stated that the first branch is related to flexural vibrations and waves, whereas the second branch to shear vibrations and waves. Since this is about the ratio between the bending and shear components, which are present in both spectra, the difference between the two spectra appears to be purely quantitative. Some authors cast doubt on the physical sense of the second spectrum for both Timoshenko beams and for plates with shear compliance and proposed to reject the associated results [7, 17, 18]. A literature review indicates that there has yet been no consensus on the sense of the second eigenvalue spectrum. An important fact is pointed out in [19]: for the first spectrum of a Timoshenko beam, the shear and bending angles have the same phase and are added to produce the total deflection angle, whereas for the second spectrum, the shear and bending angles are opposite in phase, and the total angle is equal to their difference. This observation for beams was later confirmed in [9, 17]. As far as we know, such statements, even as illustrative calculations, have not been made for plates. Here we will use dimensionless equations of motion of a plate (in first-order shear deformation theory) that incorporate only one generalized dimensionless shear compliance parameter to rigorously substantiate the qualitative difference between the two natural frequency spectra (for a hinged plate as an example) and carry out a generalized parametrical analysis. 1. Problem Formulation. Basic Equations. There are various ways of deriving the basic equations of the first-order shear deformation theory of plates (discussed in [2, 3, 20, etc.]). The simplest method is given below. Consider an isotropic plate of constant thickness h with the coordinates x and y in the midsurface and the coordinate z running along the normal. Let w x y ( , ) denote the total deflection (resulting from bending and shear), x and y the angles of rotation of normal sections due to bending, and x and y the shear angles (averaged over the thickness of the plate). The total angles of rotation of the normal to the mid-surface in two planes are given by International Applied Mechanics, Vol. 50, No. 2, March, 2014

Functionally Graded Materials are composites, usually of ceramic and metal, in which the volume fraction of their constituents varies smoothly along an interest direction. These materials were initially proposed as a solution for thermal barriers development, but are currently used in different applications. The composition variation results in a gradual change in their properties, enhancing the performance of structures subjected to thermal and mechanical loads, but making structural analysis more complex. Plates, on the other hand, are three-dimensional flat structures in which the thickness is much smaller than their other two dimensions. Different theories have been proposed for structural analysis of plates. These theories can be grouped based on the considerations adopted for the behavior of transverse shear deformations. In this regard, there are the Kirchoff-Love Theory (also known as the Classical Plate Theory - CPT), which disregards the effect of these deformations, the Reissner-Mindlin Theory (also known as the First-Order Shear Deformation Theory - FSDT), which considers shear deformations constant throughout the plate thickness, and Higher-Order Shear Deformation Theories (HSDT’s), which consider a nonlinear variation of shear deformations through different approaches. Among these theories, the most robust and accurate ones are evidently the HSDT’s. However, it is important to clarify that they require the displacement field to have a C1 continuity, which is complex for isoparametric finite elements. Thus, Isogeometric Analysis emerges as a more viable alternative by employing high continuity elements based on Non-Uniform Rational B-Splines (NURBS) functions. Therefore, this paper presents a NURBS-based isogeometric formulation for buckling and free vibration analyses of functionally graded plates using a HSDTs. A series of tests were conducted to assess the accuracy of this formulation considering examples available in the literature with different geometries, boundary conditions, and materials. The obtained results present excellent agreement with the reference solutions for thin and thick plates.

Free vibrations of rectangular plates with interior point supports

On a nonlinear theory of plates and shells including consistent and inconsistent kinematics and the finite element method

This paper describes the generalized finite difference method to solve second-order partial differential equation systems and fourth-order partial differential equations. This method is applied to solve the problem of thin and thick elastic plates.

Geometrically nonlinear large deformation analysis of functionally graded carbon nanotube reinforced composite straight-sided quadrilateral plates

Natural frequencies of thin rectangular plates using homotopy-perturbation method

A meshless study of antisymmetric angle-ply laminates using high-order shear deformation theories